Interestingly, and most importantly for our analysis, it has been
proved in [1]} that the pressure function \(r^4p\)  of hairy matter fields
which satisfy the energy conditions (REF ) and (REF )
is non-positive and monotonically decreasing in the interval
\(r\in [r_{\text{H}},r_{\gamma }]\) :
\(\lbrace p(r)\le 0\ \ \ \text{and}\ \ \ (r^4p)^{\prime }\le 0\rbrace \ \ \ \ \text{for}\ \ \ \ r_{\text{H}}\le r\le r_{\gamma }\ .\) 
